Informational alerts are useful, even though you rarely act on them. I get a mail every few minutes from a script on my web server (it gets archived automatically). In case the server explodes, I can see the status of that application just before it went down, that is when the informational alert will become an actionable alert.
My conclusion is, use many alerts, you never know when they become useful. But use the filter functions of your email client (auto-archive in Gmail for instance) even more.
You should have your logs sent to an accessible location, be it an specific email address or location.
Having the errors sent to your standard email address just clogs up your own inbox, and causes you to ignore serious errors of the kind "The server is about to go down" until you get a phone call telling you the server is down.
My conclusion is, use many alerts, you never know when they become useful. But use the filter functions of your email client (auto-archive in Gmail for instance) even more.